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D.4.8.20 Tor
Procedure from library homolog.lib (see homolog_lib).
- Compute:
- a presentation of Tor_k(M',N'), for k=v[1],v[2],... , where
M'=coker(M) and N'=coker(N): let
| | 0 <-- M' <-- G0 <-M-- G1
0 <-- N' <-- F0 <--N- F1 <-- F2 <--...
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be a presentation of M', resp. a free resolution of N', and consider
the commutative diagram
| | 0 0 0
|^ |^ |^
Tensor(M',Fk+1) -Ak+1-> Tensor(M',Fk) -Ak-> Tensor(M',Fk-1)
|^ |^ |^
Tensor(G0,Fk+1) -Ak+1-> Tensor(G0,Fk) -Ak-> Tensor(G0,Fk-1)
|^ |^
|C |B
Tensor(G1,Fk) ----> Tensor(G1,Fk-1)
(Ak,Ak+1 induced by N and B,C induced by M).
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Let K=modulo(Ak,B), J=module(C)+module(Ak+1) and Tor=modulo(K,J),
then we have exact sequences
| | R^p --K-> Tensor(G0,Fk) --Ak-> Tensor(G0,Fk-1)/im(B),
R^q -Tor-> R^p --K-> Tensor(G0,Fk)/(im(C)+im(Ak+1)).
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Hence, Tor presents Tor_k(M',N').
- Return:
- - if v is of type int: module Tor, a presentation of Tor_k(M',N');
- if v is of type intvec: a list of Tor_k(M',N') (k=v[1],v[2],...);
- in case of a third argument of any type: list l with
| | l[1] = module Tor/list of Tor_k(M',N'),
l[2] = SB of Tor/list of SB of Tor_k(M',N'),
l[3] = matrix/list of matrices, each representing a kbase of Tor_k(M',N')
(if finite dimensional), or 0.
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- Display:
- printlevel >=0: (affine) dimension of Tor_k for each k (default).
printlevel >=1: matrices Ak, Ak+1 and kbase of Tor_k in Tensor(G0,Fk)
(if finite dimensional).
- Note:
- In order to compute Tor_k(M,N) use the command Tor(k,syz(M),syz(N));
or: list P=mres(M,2); list Q=mres(N,2); Tor(k,P[2],Q[2]);
Example:
| | LIB "homolog.lib";
int p = printlevel;
printlevel = 1;
ring r = 0,(x,y),dp;
ideal i = x2,y;
ideal j = x;
list E = Tor(0..2,i,j); // Tor_k(r/i,r/j) for k=0,1,2 over r
==> // dimension of Tor_0: 0
==> // vdim of Tor_0: 1
==>
==> // Computing Tor_1 (help Tor; gives an explanation):
==> // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M),
==> // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of
==> // coker(N), then Tensor(G0,F1)-->Tensor(G0,F0) is given by:
==> x
==> // and Tensor(G0,F2) + Tensor(G1,F1)-->Tensor(G0,F1) is given by:
==> 0,x2,y
==>
==> // dimension of Tor_1: 0
==> // vdim of Tor_1: 1
==>
==> // Computing Tor_2 (help Tor; gives an explanation):
==> // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M),
==> // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of
==> // coker(N), then Tensor(G0,F2)-->Tensor(G0,F1) is given by:
==> 0
==> // and Tensor(G0,F3) + Tensor(G1,F2)-->Tensor(G0,F2) is given by:
==> 1,x2,y
==>
==> // dimension of Tor_2: -1
==>
qring R = std(i);
ideal j = fetch(r,j);
module M = [x,0],[0,x];
printlevel = 2;
module E1 = Tor(1,M,j); // Tor_1(R^2/M,R/j) over R=r/i
==> // Computing Tor_1 (help Tor; gives an explanation):
==> // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M),
==> // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of
==> // coker(N), then Tensor(G0,F1)-->Tensor(G0,F0) is given by:
==> x,0,
==> 0,x
==> // and Tensor(G0,F2) + Tensor(G1,F1)-->Tensor(G0,F1) is given by:
==> x,0,x,0,
==> 0,x,0,x
==>
==> // dimension of Tor_1: 0
==> // vdim of Tor_1: 2
==>
list l = Tor(3,M,M,1); // Tor_3(R^2/M,R^2/M) over R=r/i
==> // Computing Tor_3 (help Tor; gives an explanation):
==> // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M),
==> // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of
==> // coker(N), then Tensor(G0,F3)-->Tensor(G0,F2) is given by:
==> x,0,0,0,
==> 0,x,0,0,
==> 0,0,x,0,
==> 0,0,0,x
==> // and Tensor(G0,F4) + Tensor(G1,F3)-->Tensor(G0,F3) is given by:
==> x,0,0,0,x,0,0,0,
==> 0,x,0,0,0,x,0,0,
==> 0,0,x,0,0,0,x,0,
==> 0,0,0,x,0,0,0,x
==>
==> // dimension of Tor_3: 0
==> // vdim of Tor_3: 4
==>
==> // columns of matrix are kbase of Tor_3 in Tensor(G0,F3)
==> 1,0,0,0,
==> 0,1,0,0,
==> 0,0,1,0,
==> 0,0,0,1
==>
printlevel = p;
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